The main contribution of this paper is to provide the first step for semi-parametric estimation of features of the generation time: Consistent density estimation. To this end we extend the methodology of [1]}, [2]} in two ways: Firstly, we estimate the densities of the incubation period and the generation time in one step with the same data and, secondly, we provide a semi-parametric framework which makes no a-priori assumptions about the incubation and generation times. We pursue this by constructing a sieve estimator based on Laguerre polynomials highlighting the flexibility of Laguerre polynomials as approximating functions. In order to identify the model we need to specify a model for the relation between the exposure window \([0,W]\)  and the infection time \(T_1\)  but we will not make parametric assumptions about the distributions of the incubation period and the generation time. Thus, we will do semi-parametric estimation.
